Hotpoint WM spin balance
My Mum had a WM52 which did the same as yours; on rinse spins if it didn't balance first time it would abort the spin and go into the next rinse without spinning, on the final spin it would attempt a few times before giving up.
A rather annoying system which (although it had to be very unbalanced to trigger it) could impede rinsing results. My Mum's machine was ok at first, but by the time it was about 2/3 years old the suspension was totally shot and it was frequently very out of balance, resulting in it often carrying out entire programmes without spinning until final spin!
Luckily the machine only lasted into it's 4th year before it packed in completely so we didn't have to live with it for too long!
Matt